Early in the movie, Strickland states he brought the creature back from the Amazon, where the natives worshiped it like a god. Later, as Elisa and Giles are attempting to take the creature from the lab, Hoffstetler, while helping them, says they need to add salt to his water along with the water conditioning chemicals he gives them in a box. However, the Amazon is a freshwater river, and if the creature was from from there, it would not be used to nor need a saltwater environment.
When Elisa floods the bathroom with water, the floor would not have been able to hold the weight and would have collapsed long before it filled as much as it did.
The general's ribbon bar is upside down. The Silver Star and Distinguished Service Medal were his highest honors, and should be on the top row, not the bottom. His WW2 Victory Medal is much lower in precedence and should be on a lower row, not the top.
T-8 fluorescent lamps were not available until 1981.
Giles (Jenkins) is fumbling to undo his seat belt in a delivery van which did not have seat belts in that time period.
When Elisa and Zelda are cleaning the men's room and Strickland comes in, you see cleaning products on a shelf. The bottle of Clorox has the present day logo rather than the one from 1962.
When Strickland (Shannon) drives home from the dealer's with a new Cadillac, the car has a regulation Maryland license plate. A temporary cardboard plate would have been issued for the first couple of weeks of ownership.
When the Israeli device detonates, it destroys the electrical box it was attached to. There is no way that the power could be back on five minutes later.
Strickland's taser lands on Dr. Robert Hoffstetler's abdomen when he is stunned, but in the next shot, the taser is on his chest.
When Octavia is on the loading dock and dials the phone, she only dials 5 digits.
After Dr. Hoffstetler is shot in the mouth, a gaping hole is left in his cheek. We then see him down on the ground, with his mouth open and a bright floodlight shining on his face, and yet no light shows through the hole at all.
As the Strickland family watches "Dobie Gillis" on TV, the son asks if they can watch "Bonanza". The two TV series were never on opposite each other: "Bonanza" was on Saturday or Sunday (depending on the year) and "Dobie Gillis" was on Tuesday nights.
An elevated highway is shown which did not likely exist in the early 60's.
A WWII era sign is shown "Loose lips sink ships" that would unlikely be still shown in the early 60's. It would make sense for a government institution to still be using surplus posters from the war, as they are still trying to discourage the sharing of secret information.
The wall calendar Eliza uses is from the fall of 1962, yet the newsreels shown on television and the narrator's reference to "the last days of a fair Prince's reign" make it clear it is early May 1963.
A 5 star general would not exist in 1962. The last 5 star general was Omar Bradley in 1950 and would have been long since retired. The U.S. military historically has only used this rank during times of war in order to achieve parity with or even outrank allied commanders from other nations under their control in combat.
When Giles is asked to leave the diner he does so without taking the artwork, which he left on the counter.
At around 17:50, security is leaving the men's room and you can hear a door latch. when the door opens and closes. The door has a handle, not a knob. Therefore you would only hear a thump as the door hit the door jamb closing.